On the Goodwillie derivatives of the identity in structured ring spectra

TL;DR

This paper constructs a homotopy coherent operad structure on the derivatives of the identity functor in structured ring spectra, establishing their algebraic properties and equivalences with the operad defining the spectra.

Contribution

It introduces a new framework for understanding derivatives of the identity functor as algebras over an operad in spectra, with applications to structured ring spectra.

Findings

Constructed a homotopy coherent operad structure on derivatives

Proved connected $ ext{O}$-algebras have natural derivatives actions

Established equivalence between derivatives and the operad $ ext{O}$

Abstract

The aim of this paper is three-fold: (i) we construct a naturally occurring highly homotopy coherent operad structure on the derivatives of the identity functor on structured ring spectra which can be described as algebras over an operad $\mathcal{O}$ in spectra, (ii) we prove that every connected $\mathcal{O}$-algebra has a naturally occurring left action of the derivatives of the identity, and (iii) we show that there is a naturally occurring weak equivalence of highly homotopy coherent operads between the derivatives of the identity on $\mathcal{O}$-algebras and the operad $\mathcal{O}$. Along the way, we introduce the notion of $\mathbf{N}$-colored operads with levels which -- by construction -- provides a precise algebraic framework for working with and comparing highly homotopy coherent operads, operads, and their algebras.